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quantizer - ECE 421 Sum 2010 Notes Set 8 Signal...

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ECE 421 - Sum 2010 Notes Set 8: Signal Quantization 1 INTRODUCTION In digital dignal processing we frequently use digital algorithms which compute values for discrete points in time, or for discrete points in “space”, like in digital imaging. The input to these algorithms is often the sampled data from Analog-to-Digital Converters. However, when we implement these signal processing algorithms on a DSP-chip (which is a computer) or in digital hardware, then there are other effects become very important as well. Two of these effects are signal quantization and finite-precision arithmetic effects. SIGNAL QUANTIZATION: Consider the case in which we want to implement an algorithm (like a filter) on a DSP chip. Like a computer, the DSP chip has a finite number of bits per word, both in instructions and data representation. The Analog-to-Digital Converter (ADC) trans- forms the continuum of values possessed by the analog signal into a finite number of possible values. Each analog sample is now represented by a finite number of bits, like 16 bits or 32 bits, and this introduces noise. We will call this case signal quantization noise and it is the topic of this set of Notes. FINITE-PRECISION EFFECTS: The DSP chip must implement the algorithm using computer structures such as ac- cumulator, storage, bus transfers, etc. Consider the case of multiplying two 18-bit words and storing the result in a 18-bit memory. This multiplication requires 32-bits for full accuracy, so we would need a 32-bit memory. If we must store the multipli- cation result in a 18-bit memory this means we have to reduce the precision of the product back to 18-bits, and this also introduces noise. This phenomena is studied in the next set of Notes These are the two quantization processes we will study in this course. We next need to quantify the concept of a digital word having a finite number of bits.
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ECE 421 - Sum 2010 Notes Set 8: Signal Quantization 2 DIGITAL WORDS Many DSP chips may use 18-bit words or 32-bit words, or even longer words. However, for simplicity we will often use 3 or 4 bit words in our examples. This will make the concepts easier to understand but the principles will extend to any wordlength. SIGN-MAGNITUDE FORMAT : Many digital devices use a 2’s-complement number representation for actual computations. However, the sign-magnitude is perhaps easier for us to understand when we are learning concepts. We will refer to algorithms implemented using “ b -bit arithmetic”, where b is the number of bits excluding the sign-bit. As an example of using this convention, the representation of two b = 4-bit data words is shown below: Sign-Magnitude Decimal Integer 0 1 0 0 0 0 5000 8 0 0 1 1 1 0 4375 7 1 0 1 1 1 0 4375 7 The in the above is the “binary point”. It is not a part of the data representation, but is shown for our benefit. The sign bit is to the left of the binary point: a 0 in the sign bit implies positive and 1 implies negative. Note that the digital words above actually require 5 bits to store in memory, but we will refer to this as 4-bit (arithmetic) words. The magnitude
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